#### Volume 22, issue 1 (2018)

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Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

### Antonio Alarcón and Ildefonso Castro-Infantes

Geometry & Topology 22 (2018) 571–590
##### Abstract

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to {ℝ}^{n}$ with $\stackrel{̄}{X\left(M\right)}={ℝ}^{n}$ forms a dense subset in the space of all conformal minimal immersions $M\to {ℝ}^{n}$ endowed with the compact-open topology. Moreover, we show that every domain in ${ℝ}^{n}$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.

Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in ${ℝ}^{n}\phantom{\rule{1em}{0ex}}\left(n\ge 3\right)$, complex curves in ${ℂ}^{n}\phantom{\rule{1em}{0ex}}\left(n\ge 2\right)$, holomorphic null curves in ${ℂ}^{n}\phantom{\rule{1em}{0ex}}\left(n\ge 3\right)$, and holomorphic Legendrian curves in ${ℂ}^{2n+1}\phantom{\rule{1em}{0ex}}\left(n\in ℕ\right)$.

##### Keywords
complete minimal surface, Riemann surface, holomorphic curve
Primary: 49Q05
Secondary: 32H02
##### Publication
Accepted: 28 April 2017
Published: 31 October 2017
Proposed: Tobias H Colding
Seconded: Dmitri Burago, Leonid Polterovich